Subgroups — #41011580

Order 27008.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₆₈₈	C₂	1	1	yes	maximal, characteristic
2	#27212972	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₂×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₈₄₄	—	2	1	no	—
5	Dic₁₆₈₈	—	2	1	no	—
6	Dic₁₆₈₈	C₄	1	2	yes	complemented
7	#20269911	—	2	1	no	—
8	C₂×C₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
9	C₃₂×C₂₁₁	—	2	1	no	—
10	C₂×Dic₄₂₂	—	4	1	no	—
11	Dic₈₄₄	—	2	2	no	—
12	Dic₈₄₄	—	4	1	no	—
13	C₄⋊Dic₂₁₁	—	4	1	no	—
14	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
15	C₁₆×C₂₁₁	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₂₁₁	D₄	1	1	yes	characteristic
17	Dic₄₂₂	—	4	2	no	—
18	Dic₄₂₂	—	8	1	no	—
19	C₂×Dic₂₁₁	—	8	1	no	—
20	C₂×Dic₂₁₁	—	8	1	no	—
21	C₂×C₄×C₂₁₁	D₈	1	1	yes	characteristic
22	C₈×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₂₁₁	SD₁₆	1	1	yes	characteristic
24	C₂²×C₂₁₁	D₁₆	1	1	yes	characteristic
25	C₄×C₂₁₁	#102	1	1	yes	characteristic
26	C₄×C₂₁₁	SD₃₂	1	1	yes	characteristic
27	Dic₂₁₁	—	8	2	no	—
28	Dic₂₁₁	—	16	1	no	—
29	C₂×C₂₁₁	#357	1	1	yes	characteristic
30	C₂×C₂₁₁	SD₆₄	1	1	yes	characteristic
31	C₂×C₂₁₁	Q₆₄	1	1	yes	characteristic
32	C₂₁₁	#1416	1	1	yes	characteristic, complemented, Sylow
33	#1416	—	211	1	no	maximal, Sylow
34	C₂×Q₃₂	—	211	1	no	—
35	C₂×C₃₂	D₂₁₁	1	1	yes	characteristic
36	#366	—	211	1	no	—
37	C₂×Q₁₆	—	422	1	no	—
38	Q₃₂	—	422	1	no	—
39	Q₃₂	—	211	2	no	—
40	C₂×C₁₆	C₂×D₂₁₁	1	1	yes	characteristic
41	#107	—	422	1	no	—
42	C₃₂	—	2	1	no	—
43	C₂×Q₈	—	844	1	no	—
44	Q₁₆	—	844	1	no	—
45	Q₁₆	—	422	2	no	—
46	C₂×C₈	D₈₄₄	1	1	yes	characteristic
47	C₄⋊C₄	—	844	1	no	—
48	C₁₆	C₄×D₂₁₁	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₂₁₁	1	1	yes	characteristic
50	Q₈	—	844	2	no	—
51	Q₈	—	1688	1	no	—
52	C₂×C₄	—	1688	1	no	—
53	C₂×C₄	—	1688	1	no	—
54	C₂×C₄	D₁₆₈₈	1	1	yes	characteristic
55	C₈	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
56	C₈	#16809951	1	1	yes	characteristic
57	C₂²	D₃₃₇₆	1	1	yes	characteristic
58	C₄	—	3376	1	no	—
59	C₄	—	1688	2	no	—
60	C₄	#20269899	1	1	yes	characteristic
61	C₄	#20269922	1	1	yes	characteristic
62	C₂	#27212948	1	1	yes	characteristic
63	C₂	#27212985	1	1	yes	characteristic
64	C₂	Dic₃₃₇₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃₃₇₆
commutator	C₁₆×C₂₁₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₁₁
Fitting	C₂×C₃₂×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	D₁₆
cosocle	C₂×C₁₆×C₂₁₁	C₂²
hypercenter	C₂×C₃₂	D₂₁₁
hypocenter	C₂₁₁	#1416
LCS	G ▷ C₁₆×C₂₁₁ ▷ C₈×C₂₁₁ ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
6. Dic₁₆₈₈	C₄	semidirect
32. C₂₁₁	#1416	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×Dic₁₆₈₈	2
#27212972	2
C₂×C₃₂×C₂₁₁	2
#1416	211